API

Result object returned by GRAPE.optimize.

Attributes

The attributes of a GrapeResult object include

• iter: The number of the current iteration
• J_T: The value of the final-time functional in the current iteration
• J_T_prev: The value of the final-time functional in the previous iteration
• J_a: The value of the running cost $J_a$ in the current iteration (excluding $λ_a$)
• J_a_prev: The value of $J_a$ in the previous iteration
• tlist: The time grid on which the control are discetized.
• guess_controls: A vector of the original control fields (each field discretized to the points of tlist)
• optimized_controls: A vector of the optimized control fields in the current iterations
• records: A vector of tuples with values returned by a callback routine passed to optimize
• converged: A boolean flag on whether the optimization is converged. This may be set to true by a check_convergence function.
• message: A message string to explain the reason for convergence. This may be set by a check_convergence function.

All of the above attributes may be referenced in a check_convergence function passed to QuantumControl.optimize(problem; method=GRAPE) or GRAPE.optimize.

GRAPE Workspace.

The workspace is for internal use. However, it is also accessible in a callback function. The callback may use or modify some of the following attributes:

• trajectories: a copy of the trajectories defining the control problem
• tlist: the time grid for the optimization
• adjoint_trajectories: The trajectories with the adjoint generator
• kwargs: The keyword arguments from the call to optimize.
• controls: A tuple of the original controls (probably functions)
• pulsevals_guess: The combined vector of pulse values that are the guess in the current iteration. Initially, the vector is the concatenation of discretizing controls to the midpoints of the time grid.
• pulsevals: The combined vector of updated pulse values in the current iteration. All the initialized propagators inside the workspace alias pulsevals such that mutating pulsevals is directly reflected in the next propagation step.
• gradient: The total gradient for the guess in the current iteration
• grad_J_T: The current gradient for the final-time part of the functional. This is from the last evaluation of the gradient, which may be for the optimized pulse (depending on the internal of the optimizer)
• grad_J_a: The current gradient for the running cost part of the functional.
• J_parts: The two-component vector $[J_T, J_a]$
• upper_bounds: Upper bound for every pulsevals; +Inf indicates no bound.
• lower_bounds: Lower bound for every pulsevals; -Inf indicates no bound.
• fg_count: A two-element vector containing the number of evaluations of the combined gradient and functional first, and the evaluations of only the functional second.
• optimizer: The backend optimizer object
• optimizer_state: The internal state object of the optimizer (nothing if the optimizer has no internal state)
• result: The current result object
• tau_grads: The gradients ∂τₖ/ϵₗ(tₙ)
• fw_storage: The storage of states for the forward propagation, as a vector of storage contains (one for each trajectory)
• fw_propagators: The propagators used for the forward propagation
• bw_grad_propagators: The propagators used for the backward propagation of QuantumGradientGenerators.GradVector states (gradient_method=:gradgen only)
• bw_propagators: The propagators used for the backward propagation (gradient_method=:taylor only)
• use_threads: Flag indicating whether the propagations are performed in parallel.

In addition, the following methods provide safer (non-mutating) access to information in the workspace

• step_width
• search_direction
• norm_search
• gradient
• pulse_update

Evaluate the optimization functional encoded in wrk for the given pulsevals.

J = evaluate_functional(pulsevals, wrk; storage=nothing, count_call=true)

evaluates the functional defined during the initialization of the GRAPE workspace wrk, for the given pulse values, using wrk.fw_propagators. The pulsevals argument is a vector of Float64 values corresponding to a concatenation of all the controls, discretized to the midpoints of the time grid, cf. GrapeWrk.

As a side effect, the evaluation sets the following information in wrk:

• wrk.pulsevals: On output, the values of the given pulsevals. Note that pulsevals may alias wrk.pulsevals, so there is no assumption made on wrk.pulsevals other than that mutating wrk.pulsevals directly affects the propagators in wrk.
• wrk.result.f_calls: Will be incremented by one (only if count_call=true)
• wrk.fg_count[2]: Will be incremented by one (only if count_call=true)
• wrk.result.tau_vals: For any trajectory that defines a target_state, the overlap of the propagated state with that target state.
• wrk.J_parts: The parts (J_T, λₐJ_a) of the functional

If storage is given, as a vector of storage containers suitable for propagate (one for each trajectory), the forward-propagated states will be stored there.

Returns J as sum(wrk.J_parts).

Evaluate the gradient $∂J/∂ϵₙₗ$ into G, together with the functional J.

J = evaluate_gradient!(G, pulsevals, wrk)

evaluates and returns the optimization functional defined during the initialization of wrk, for the given pulse values, cf. evaluate_functional, and write the derivative of the optimization functional with respect to the pulse values into the existing array G.

The evaluation of the functional uses uses wrk.fw_propagators. The evaluation of the gradient happens either via a backward propagation of an extended "gradient vector" using wrk.bw_grad_propagators if wrk was initialized with gradient_method=:gradgen. Alternatively, if wrk was initialized with gradient_method=:taylor, the backward propagation if for a regular state, using wrk.bw_propagators, and a Taylor expansion is used for the gradient of the time evolution operator in a single time step.

As a side, effect, evaluating the gradient and functional sets the following information in wrk:

• wrk.pulsevals: On output, the values of the given pulsevals, see evaluate_functional.
• wrk.result.fg_calls: Will be incremented by one
• wrk.fg_count[1]: Will be incremented by one
• wrk.result.tau_vals: For any trajectory that defines a target_state, the overlap of the propagated state with that target state.
• wrk.J_parts: The parts (J_T, λₐJ_a) of the functional
• wrk.fw_storage: For each trajectory, the forward-propagated states at each point on the time grid.
• wrk.chi_states: The normalized states $|χ(T)⟩$ that we used as the boundary condition for the backward propagation.
• wrk.chi_states_norm: The original norm of the states $|χ(T)⟩$, as calculated by $-∂J/∂⟨Ψₖ|$
• wrk.grad_J_T: The vector ``∂JT/∂ϵ{nl}, i.e., the gradient only for the final-time part of the functional
• wrk.grad_J_a: The vector $∂J_a/∂ϵ_{nl}$, i.e., the gradient only for the pulse-dependent running cost.

The gradients are wrk.grad_J_T and wrk.grad_J_a (weighted by $λ_a$) into are combined into the output G.

Returns the value of the functional.

The gradient in the current iteration.

g = gradient(wrk; which=:initial)

returns the gradient associated with the guess pulse of the current iteration. Up to quasi-Newton corrections, the negative gradient determines the search_direction for the pulse_update.

g = gradient(wrk; which=:final)

returns the gradient associated with the optimized pulse of the current iteration.

Print optimization progress as a table.

print_iters = make_grape_print_iters(; print_iter_info, store_iter_info=[])

generates a print_iters function that can be passed as callback to GRAPE.optimize. It is also used automatically when GRAPE.optimized is called via QuantumControl.optimize with print_iters=true.

The print_iter_info keyword argument specifies what information should be printed, and defaults to ["iter.", "J_T", "|∇J|", "|Δϵ|", "ΔJ", "FG(F)", "secs"]. The store_iter_info similarly specifies what information should be returned from the callback, so that it can be stored in the records field of the GrapeResult object.

The available fields for print_iter_info and store_iter_info are:

• "iter.": The iteration number
• "J_T": The value of the final-time functional for the dynamics under the optimized pulses
• "J_a": The value of the pulse-dependent running cost for the optimized pulses
• "λ_a⋅J_a": The total contribution of J_a to the full functional J
• "J": The value of the optimization functional for the optimized pulses
• "ǁ∇J_Tǁ": The ℓ²-norm of the current gradient of the final-time functional. Note that this is usually the gradient of the optimize pulse, not the guess pulse.
• "ǁ∇J_aǁ": The ℓ²-norm of the the current gradient of the pulse-dependent running cost. For comparison with "ǁ∇J_Tǁ".
• "λ_aǁ∇J_aǁ": The ℓ²-norm of the the current gradient of the complete pulse-dependent running cost term. For comparison with "ǁ∇J_Tǁ".
• "ǁ∇Jǁ": The norm of the guess pulse gradient. Note that the guess pulse gradient is not the same the current gradient.
• "ǁΔϵǁ": The ℓ²-norm of the pulse update
• "ǁϵǁ": The ℓ²-norm of optimized pulse values
• "max|Δϵ|" The maximum value of the pulse update (infinity norm)
• "max|ϵ|": The maximum value of the pulse values (infinity norm)
• "ǁΔϵǁ/ǁϵǁ": The ratio of the pulse update tothe optimized pulse values
• "∫Δϵ²dt": The L²-norm of the pulse update, summed over all pulses. A convergence measure comparable (proportional) to the running cost in Krotov's method
• "ǁsǁ": The norm of the search direction. Should be ǁΔϵǁ scaled by the step with α.
• "∠°": The angle (in degrees) between the negative gradient -∇J and the search direction s.
• "α": The step width as determined by the line search (Δϵ = α⋅s)
• "ΔJ_T": The change in the final time functional relative to the previous iteration
• "ΔJ_a": The change in the control-dependent running cost relative to the previous iteration
• "λ_a⋅ΔJ_a": The change in the control-dependent running cost term relative to the previous iteration.
• "ΔJ": The change in the total optimization functional relative to the previous iteration.
• "FG(F)": The number of functional/gradient evaluation (FG), or pure functional (F) evaluations
• "secs": The number of seconds of wallclock time spent on the iteration.

The norm of the search direction vector in the current iteration.

norm_search(wrk)

returns norm(search_direction(wrk)).

Solve a quantum control problem using the GRAPE method.

using GRAPE
result = GRAPE.optimize(trajectories, tlist; J_T, kwargs...)

minimizes a functional

\[J(\{ϵ_{nl}\}) = J_T(\{|Ψ_k(T)⟩\}) + λ_a J_a(\{ϵ_{nl}\})\,,\]

via the GRAPE method, where the final time functional $J_T$ depends explicitly on the forward-propagated states $|Ψ_k(T)⟩$, where $|Ψ_k(t)⟩$ is the time evolution of the initial_state in the $k$th' element of the trajectories, and the running cost $J_a$ depends explicitly on pulse values $ϵ_{nl}$ of the l'th control discretized on the n'th interval of the time grid tlist.

It does this by calculating the gradient of the final-time functional

\[\nabla J_T \equiv \frac{\partial J_T}{\partial ϵ_{nl}} = -2 \Re \underbrace{% \underbrace{\bigg\langle χ(T) \bigg\vert \hat{U}^{(k)}_{N_T} \dots \hat{U}^{(k)}_{n+1} \bigg \vert}_{\equiv \bra{\chi(t_n)}\;\text{(bw. prop.)}} \frac{\partial \hat{U}^{(k)}_n}{\partial ϵ_{nl}} }_{\equiv \bra{χ_k^\prime(t_{n-1})}} \underbrace{\bigg \vert \hat{U}^{(k)}_{n-1} \dots \hat{U}^{(k)}_1 \bigg\vert Ψ_k(t=0) \bigg\rangle}_{\equiv |\Psi(t_{n-1})⟩\;\text{(fw. prop.)}}\,,\]

where $\hat{U}^{(k)}_n$ is the time evolution operator for the $n$ the interval, generally assumed to be $\hat{U}^{(k)}_n = \exp[-i \hat{H}_{kn} dt_n]$, where $\hat{H}_{kn}$ is the operator obtained by evaluating trajectories[k].generator on the $n$'th time interval.

The backward-propagation of $|\chi_k(t)⟩$ has the boundary condition

\[ |\chi_k(T)⟩ \equiv - \frac{\partial J_T}{\partial ⟨\Psi_k(T)|}\,.\]

The final-time gradient $\nabla J_T$ is combined with the gradient for the running costs, and the total gradient is then fed into an optimizer (L-BFGS-B by default) that iteratively changes the values $\{ϵ_{nl}\}$ to minimize $J$.

See Background for details.

Returns a GrapeResult.

Positional arguments

• trajectories: A vector of Trajectory objects. Each trajectory contains an initial_state and a dynamical generator (e.g., time-dependent Hamiltonian). Each trajectory may also contain arbitrary additional attributes like target_state to be used in the J_T functional
• tlist: A vector of time grid values.

Required keyword arguments

• J_T: A function J_T(Ψ, trajectories) that evaluates the final time functional from a list Ψ of forward-propagated states and trajectories. The function J_T may also take a keyword argument tau. If it does, a vector containing the complex overlaps of the target states (target_state property of each trajectory in trajectories) with the propagated states will be passed to J_T.

Optional keyword arguments

• chi: A function chi(Ψ, trajectories) that receives a list Ψ of the forward propagated states and returns a vector of states $|χₖ⟩ = -∂J_T/∂⟨Ψₖ|$. If not given, it will be automatically determined from J_T via QuantumControl.Functionals.make_chi with the default parameters. Similarly to J_T, if chi accepts a keyword argument tau, it will be passed a vector of complex overlaps.
• chi_min_norm=1e-100: The minimum allowable norm for any $|χₖ(T)⟩$. Smaller norms would mean that the gradient is zero, and will abort the optimization with an error.
• J_a: A function J_a(pulsevals, tlist) that evaluates running costs over the pulse values, where pulsevals are the vectorized values $ϵ_{nl}$, where n are in indices of the time intervals and l are the indices over the controls, i.e., [ϵ₁₁, ϵ₂₁, …, ϵ₁₂, ϵ₂₂, …] (the pulse values for each control are contiguous). If not given, the optimization will not include a running cost.
• gradient_method=:gradgen: One of :gradgen (default) or :taylor. With gradient_method=:gradgen, the gradient is calculated using QuantumGradientGenerators. With gradient_method=:taylor, it is evaluated via a Taylor series, see Eq. (20) in Kuprov and Rogers, J. Chem. Phys. 131, 234108 (2009) [22].
• taylor_grad_max_order=100: If given with gradient_method=:taylor, the maximum number of terms in the Taylor series. If taylor_grad_check_convergence=true (default), if the Taylor series does not convergence within the given number of terms, throw an an error. With taylor_grad_check_convergence=true, this is the exact order of the Taylor series.
• taylor_grad_tolerance=1e-16: If given with gradient_method=:taylor and taylor_grad_check_convergence=true, stop the Taylor series when the norm of the term falls below the given tolerance. Ignored if taylor_grad_check_convergence=false.
• taylor_grad_check_convergence=true: If given as true (default), check the convergence after each term in the Taylor series an stop as soon as the norm of the term drops below the given number. If false, stop after exactly taylor_grad_max_order terms.
• lambda_a=1: A weight for the running cost J_a.
• grad_J_a: A function to calculate the gradient of J_a. If not given, it will be automatically determined. See make_grad_J_a for the required interface.
• upper_bound: An upper bound for the value of any optimized control. Time-dependent upper bounds can be specified via pulse_options.
• lower_bound: A lower bound for the value of any optimized control. Time-dependent lower bounds can be specified via pulse_options.
• pulse_options: A dictionary that maps every control (as obtained by get_controls from the trajectories) to a dict with the following possible keys:
  • :upper_bounds: A vector of upper bound values, one for each intervals of the time grid. Values of Inf indicate an unconstrained upper bound for that time interval, respectively the global upper_bound, if given.
  • :lower_bounds: A vector of lower bound values. Values of -Inf indicate an unconstrained lower bound for that time interval,
• callback: A function that receives the GRAPE workspace and the iteration number. The function may return a tuple of values which are stored in the GrapeResult object result.records. The function can also mutate the workspace, in particular the updated pulsevals. This may be used, e.g., to apply a spectral filter to the updated pulses or to perform similar manipulations.
• check_convergence: A function to check whether convergence has been reached. Receives a GrapeResult object result, and must return one of the following:
  • A boolean (true if convergence is reached, false otherwise)
  • A string with a reason for the convergence, or an empty string if not converged.
  • The original result object or nothing, indicating that result.converged and result.message may have been modified to indicate convergence
  The convergence check is performed after any callback.
• prop_method: The propagation method to use for each trajectory, see below.
• verbose=false: If true, print information during initialization
• rethrow_exceptions: By default, any exception ends the optimization, but still returns a GrapeResult that captures the message associated with the exception. This is to avoid losing results from a long-running optimization when an exception occurs in a later iteration. If rethrow_exceptions=true, instead of capturing the exception, it will be thrown normally.

Experimental keyword arguments

The following keyword arguments may change in non-breaking releases:

• x_tol: Parameter for Optim.jl
• f_tol: Parameter for Optim.jl
• g_tol: Parameter for Optim.jl
• show_trace: Parameter for Optim.jl
• extended_trace: Parameter for Optim.jl
• show_every: Parameter for Optim.jl
• allow_f_increases: Parameter for Optim.jl
• optimizer: An optional Optim.jl optimizer (Optim.AbstractOptimizer instance). If not given, an L-BFGS-B optimizer will be used.

Trajectory propagation

GRAPE may involve three types of time propagation, all of which are implemented via the QuantumPropagators as a numerical backend:

• A forward propagation for every Trajectory in the trajectories
• A backward propagation for every trajectory
• A backward propagation of a gradient generator for every trajectory.

The keyword arguments for each propagation (see propagate) are determined from any properties of each Trajectory that have a prop_ prefix, cf. init_prop_trajectory.

In situations where different parameters are required for the forward and backward propagation, instead of the prop_ prefix, the fw_prop_ and bw_prop_ prefix can be used, respectively. These override any setting with the prop_ prefix. Similarly, properties for the backward propagation of the gradient generators can be set with properties that have a grad_prop_ prefix. These prefixes apply both to the properties of each Trajectory and the keyword arguments.

Note that the propagation method for each propagation must be specified. In most cases, it is sufficient (and recommended) to pass a global prop_method keyword argument.

The vector of pulse update values for the current iteration.

Δu = pulse_update(wrk)

returns a vector containing the different between the optimized pulse values and the guess pulse values of the current iteration. This should be proportional to search_direction with the proportionality factor step_width.

The search direction used in the current iteration.

s = search_direction(wrk)

returns the vector describing the search direction used in the current iteration. This should be proportional to pulse_update with the proportionality factor step_width.

The step width used in the current iteration.

α = step_width(wrk)

returns the scalar α so that pulse_update(wrk) = α * search_direction(wrk), see pulse_update and search_direction for the iteration described by the current GrapeWrk (for the state of wrk as available in the callback of the current iteration.

The angle between two vectors.

ϕ = vec_angle(v1, v2; unit=:rad)

returns the angle between two vectors in radians (or degrees, with unit=:degree).

using GRAPE
result = optimize(problem; method=GRAPE, kwargs...)

optimizes the given QuantumControl.ControlProblem using the GRAPE (Gradient-Ascent Pulse Engineering) method.

Delegates to

result = GRAPE.optimize(
    problem.trajectories, problem.tlist; problem.kwargs..., kwargs...
)

See GRAPE.optimize for details and supported keyword arguments.

Compared to calling GRAPE.optimize directly, the QuantumControl.optimize wrapper adds the following additional keyword arguments:

• check=true: If true (default), test that all the objects stored in the trajectories implement the required interfaces correctly
• print_iters=true: Whether to print information after each iteration.
• print_iter_info=["iter.", "J_T", "|∇J|", "|Δϵ|", "ΔJ", "FG(F)", "secs"]: Which fields to print if print_iters=true. See make_grape_print_iters
• store_iter_info=[]: Which fields to store in result.records, given as a list of header labels, see print_iter_info. See make_grape_print_iters

These options still allow for the normal callback argument. With QuantumcControl.optimize, the callback can be a tuple of callback functions that will be combined automatically, which GRAPE.optimize only supports as single callback function.

The GRAPE optimization may also be initiated via QuantumControl.@optimize_or_load, which additionally adds checkpointing, to ensure that an optimization result is dumped to disk in case of an unexpected shutdown.

Description of a state's time evolution.

Trajectory(
    initial_state,
    generator;
    target_state=nothing,
    weight=1.0,
    kwargs...
)

describes the time evolution of the initial_state under a time-dependent dynamical generator (e.g., a Hamiltonian or Liouvillian).

Trajectories are central to quantum control problems: an optimization functional depends on the result of propagating one or more trajectories. For example, when optimizing for a quantum gate, the optimization considers the trajectories of all logical basis states.

In addition to the initial_state and generator, a Trajectory may include data relevant to the propagation and to evaluating a particular optimization functional. Most functionals have the notion of a "target state" that the initial_state should evolve towards, which can be given as the target_state keyword argument. In some functionals, different trajectories enter with different weights [38], which can be given as a weight keyword argument. Any other keyword arguments are also available to a functional as properties of the Trajectory .

A Trajectory can also be instantiated using all keyword arguments.

Properties

All keyword arguments used in the instantiation are available as properties of the Trajectory. At a minimum, this includes initial_state, generator, target_state, and weight.

By convention, properties with a prop_ prefix, e.g., prop_method, will be taken into account when propagating the trajectory. See propagate_trajectory for details.

Set the default provider for automatic differentiation.

QuantumControl.set_default_ad_framework(mod; quiet=false)

registers the given module (package) as the default AD framework.

This determines the default setting for the automatic parameter in the following functions:

• QuantumControl.Functionals.make_chi
• QuantumControl.Functionals.make_gate_chi
• QuantumControl.Functionals.make_grad_J_a

The given mod must be a supported AD framework, e.g.,

import Zygote
QuantumControl.set_default_ad_framework(Zygote)

Currently, there is built-in support for Zygote and FiniteDifferences.

For other packages to be used as the default AD framework, the appropriate methods for make_chi etc. must be defined.

Unless quiet=true, calling set_default_ad_framework will show a message to confirm the setting.

To unset the default AD framework, use

QuantumControl.set_default_ad_framework(nothing)